ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 525 



manifest that the mechanical problem may be solved, though the reciprocal figure 

 cannot be constructed owing to the condition of all the sides of a face lying 

 in a plane not being fulfilled, or owing to a face belonging to more than two 

 cells. Hence the mechanical interest of reciprocal figures in space rapidly 

 diminishes with their complexity. 



Diagrams of forces in which the forces are represented by lines may be 

 always constructed in space as well as in a plane, but in general some of the 

 lines must be repeated. 



Thus in the figure of five points, each point is the meeting place of four 

 lines. The forces in these lines may be represented by five gauche quadrilaterals 

 (that is, quadrilaterals not in one plane) ; and one of these being chosen, the 

 other four may be applied to its sides and to each other so as to form five 

 sides of a gauche hexahedron. The sixth side, that opposite the original quad- 

 rilateral, will be a parallelogram, the opposite sides of which are repetitions of 

 the same line. 



We have thus a complete but redundant diagram of forces consisting of 

 eight points joined by twelve lines, two pairs of the lines being repetitions. 

 This is a more convenient though less elegant construction of a diagram of 

 forces, and it never becomes geometrically impossible as long as the problem is 

 mechanically possible, however complicated the original figure may be. 



